Subgroups — #36351719

Order 22144.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011881	C₂	1	1	yes	maximal, characteristic
2	#26011288	C₂	1	1	yes	maximal, characteristic, complemented
3	#354×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14583813	—	2	1	no	—
5	C₂×#14583811	—	2	1	no	—
6	#19149676	C₄	1	1	yes	characteristic, complemented
7	#19149598	—	2	1	no	—
8	#19149590	C₄	1	1	yes	characteristic, complemented
9	#124×C₁₇₃	C₂²	1	1	yes	characteristic
10	#99×C₁₇₃	—	2	1	no	—
11	C₂×D₆₉₂	—	2	1	no	—
12	C₂×D₆₉₂	—	4	1	no	—
13	#14583813	—	4	2	no	—
14	#14583811	—	4	2	no	—
15	#14583810	—	4	1	no	—
16	C₂×#13063436	—	2	1	no	—
17	C₂×Q₈×C₁₇₃	D₄	1	1	yes	characteristic
18	C₂×D₄×C₁₇₃	D₄	1	1	yes	characteristic
19	C₂²⋊C₄×C₁₇₃	—	4	1	no	—
20	C₂²⋊C₄×C₁₇₃	—	4	1	no	—
21	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₁₇₃	—	4	1	no	—
23	D₆₉₂	—	4	1	no	—
24	D₆₉₂	—	4	1	no	—
25	D₆₉₂	—	8	2	no	—
26	#13063436	—	2	2	no	—
27	#13063436	—	8	1	no	—
28	C₂³×C₁₇₃	—	2	1	no	—
29	Q₈×C₁₇₃	—	4	1	no	—
30	D₄×C₁₇₃	—	4	1	no	—
31	C₂×C₄×C₁₇₃	—	2	1	no	—
32	C₂×C₄×C₁₇₃	—	2	1	no	—
33	C₂×C₄×C₁₇₃	—	8	1	no	—
34	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₁₇₃	—	4	1	no	—
36	C₂²×C₁₇₃	—	8	1	no	—
37	C₂²×C₁₇₃	#99	1	1	yes	characteristic
38	C₂×D₁₇₃	—	8	2	no	—
39	C₂×D₁₇₃	—	8	1	no	—
40	C₄×C₁₇₃	—	2	1	no	—
41	C₄×C₁₇₃	—	4	1	no	—
42	C₄×C₁₇₃	—	4	1	no	—
43	C₄×C₁₇₃	—	16	1	no	—
44	C₂×C₁₇₃	—	2	1	no	—
45	C₂×C₁₇₃	—	8	1	no	—
46	C₂×C₁₇₃	#353	1	1	yes	characteristic
47	D₁₇₃	—	16	1	no	—
48	C₁₇₃	#1406	1	1	yes	characteristic, complemented, Sylow
49	#1406	—	173	1	no	maximal, Sylow
50	#496	—	173	1	no	—
51	#355	—	173	1	no	—
52	#354	D₁₇₃	1	1	yes	characteristic, complemented
53	C₂×SD₁₆	—	346	1	no	—
54	C₂×D₈	—	346	1	no	—
55	#127	—	173	1	no	—
56	#124	C₂×D₁₇₃	1	1	yes	characteristic
57	#101	—	346	1	no	—
58	#99	—	2	1	no	—
59	#97	—	173	1	no	—
60	C₂×Q₈	D₆₉₂	1	1	yes	characteristic
61	C₂×D₄	—	346	1	no	—
62	C₂×D₄	—	692	1	no	—
63	C₂×D₄	C₂²⋊D₁₇₃	1	1	yes	characteristic
64	SD₁₆	—	692	2	no	—
65	D₈	—	692	2	no	—
66	M₁₆	—	692	1	no	—
67	C₂×C₈	—	346	1	no	—
68	C₂²⋊C₄	—	4	1	no	—
69	C₂²⋊C₄	—	4	1	no	—
70	C₄²	C₄×D₁₇₃	1	1	yes	characteristic
71	C₂³	—	2	1	no	—
72	C₂³	—	692	1	no	—
73	Q₈	—	4	1	no	—
74	D₄	—	4	1	no	—
75	D₄	—	1384	2	no	—
76	D₄	—	692	1	no	—
77	D₄	—	692	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	2	1	no	—
80	C₂×C₄	—	8	1	no	—
81	C₂×C₄	#14583803	1	1	yes	characteristic
82	C₈	—	346	2	no	—
83	C₈	—	1384	1	no	—
84	C₂²	—	4	1	no	—
85	C₂²	—	8	1	no	—
86	C₂²	—	1384	2	no	—
87	C₂²	—	1384	1	no	—
88	C₂²	#19149593	1	1	yes	characteristic
89	C₄	—	2	1	no	—
90	C₄	—	4	1	no	—
91	C₄	—	4	1	no	—
92	C₄	—	16	1	no	—
93	C₂	—	2	1	no	—
94	C₂	—	8	1	no	—
95	C₂	—	2768	1	no	—
96	C₂	#26011283	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26011283
commutator	C₄²×C₁₇₃	C₂×C₄
Frattini	#124	C₂×D₁₇₃
Fitting	#354×C₁₇₃	C₂
socle	C₂×C₁₇₃	#353
prime radical	#26011881	C₂
cosocle	#124×C₁₇₃	C₂²
hypercenter	#354	D₁₇₃
hypocenter	C₁₇₃	#1406
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂×C₄×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #354

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #26011288	C₂	semidirect
3. #354×C₁₇₃	C₂	semidirect
6. #19149676	C₄	semidirect
8. #19149590	C₄	semidirect
48. C₁₇₃	#1406	semidirect
52. #354	D₁₇₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#26011881	2
#26011288	2
#354×C₁₇₃	2
#1406	173